# 三维各向异性重力正演模型的高阶展开项评估Evaluation on High-Degree Series Expansion of the 3-D Isotropic Forward Gravity Field Modelling

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The global gravity forward modelling of the shallow layer mass distribution is of vital importance in the computation of Bouguer anomaly, isostatic compensation and ultra-high resolution Earth’s gravity field. In this paper, based on the 3-D isotropic gravity forward modelling method, the contribution of each expansion power to the overall resultant forward modelling gravity in each geological province of GEMMA crustal model is further investigated. Consequently, based on the discrepancy of individual binominal expansion of each geological province and the overall resultant forward modelling gravity, for the accuracy criterion of 0.1 mGal, if considering only the influence of forward modelling method, the truncation degree/order of geological provinces with shallower crust is 6, while the one with thicker crust is supposed to truncate at 12th power.

1. 引言

2. 数据与方法

2.1. GEMMA地壳模型

GEMMA地壳结构模型是欧洲航天局(ESA, European Space Agency)的地球重力场和海洋环流探测卫星(GOCE, Gravity field and steady-state Ocean Circulation Explorer)的重力梯度产品的应用之一。该地壳结构模型反演地壳地幔边界Moho面与地壳密度时，以地震波折射资料反演得到的CRUST2.0\cite地壳结构模型为基础，考虑USGS1995发布的大地构造分区、全球尺度因子及地壳密度随深度变化的因素，直接使用GOCE重力梯度径向观测值进行反演。最终，联合反演得到分辨率为0.5˚ × 0.5˚的地壳结构模型GEMMA，该模型在全球范围内与GOCE重力梯度观测值有较好一致性，GEMMA正演所得对应重力观测值与GOCE重力梯度观测值总体标准差在49 mE。

2.2. 三维各向异性的重力正演

$V\left(P\right)=\frac{GM}{R}\underset{n,m}{\overset{\infty }{\sum }}{\left(\frac{R}{r}\right)}^{n+1}\frac{1}{M\left(2n+1\right)}{Y}_{nm}\left(\Omega \right){\int }_{\Sigma }{\left(\frac{{r}^{\prime }}{R}\right)}^{n}\rho \left({r}^{\prime },{\Omega }^{\prime }\right){Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Sigma$ (1)

${Y}_{nm}\left(\theta ,\lambda \right)={P}_{nm}\left(\mathrm{cos}\varphi \right)\left\{\begin{array}{ll}\mathrm{cos}\left(m\lambda \right),\hfill & m\le 0\hfill \\ \mathrm{sin}\left(m\lambda \right),\hfill & m>0\hfill \end{array}$ (2)

${V}_{nm}=\frac{3}{4\text{π}\stackrel{¯}{\rho }{R}^{3}\left(2n+1\right)}{\int }_{\Sigma }{\left(\frac{{r}^{\prime }}{R}\right)}^{n}\rho \left({r}^{\prime },{\Omega }^{\prime }\right){Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Sigma$ (3)

${V}_{nm}^{\left(\omega \right)}=\frac{3}{4\text{π}{R}^{n+3}\left(2n+1\right)}{\int }_{\sigma }{\int }_{{{R}^{\prime }}^{\left(\omega \right)}+{r}_{i}^{\left(\omega \right)}\left(\Omega \right)}^{{{R}^{\prime }}^{\left(\omega \right)}+{r}_{e}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)}{{r}^{\prime }}^{n+2}\frac{{\rho }^{\left(\omega \right)}\left({r}^{\prime },{\Omega }^{\prime }\right)}{\stackrel{¯}{\rho }}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega \text{d}r$ (4)

${\rho }^{\left(\omega \right)}\left({r}^{\prime },{\Omega }^{\prime }\right)={\rho }^{\left(\omega \right)}\left({\Omega }^{\prime }\right)+\underset{i=1}{\overset{I}{\sum }}{\alpha }_{i}\left({\Omega }^{\prime }\right){\left({R}^{\prime }-{r}^{\prime }\right)}^{i}$ (5)

$\begin{array}{c}{\int }_{{{R}^{\prime }}^{\left(\omega \right)}+{r}_{i}^{\left(\omega \right)}\left(\Omega \right)}^{{{R}^{\prime }}^{\left(\omega \right)}+{r}_{e}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)}{{r}^{\prime }}^{n+2}\text{d}r=\frac{{{R}^{\prime }}^{\left(\omega \right)n+3}}{n+3}\underset{k=0}{\overset{\sigma }{\sum }}\left(\begin{array}{c}n+3\\ k\end{array}\right)\left[{\left(\frac{{r}_{e}^{\left(\omega \right)}}{{{R}^{\prime }}^{\left(\omega \right)}}\right)}^{k}-{\left(\frac{{r}_{i}^{\left(\omega \right)}}{{{R}^{\prime }}^{\left(\omega \right)}}\right)}^{k}\right]+ϵ\sigma \\ =\frac{{{R}^{\prime }}^{\left(\omega \right)n+3}}{n+3}\underset{k=0}{\overset{\sigma }{\sum }}\left(\begin{array}{c}n+3\\ k\end{array}\right){F}_{k}^{\omega }\left({\Omega }^{\prime }\right)+ϵ\sigma \end{array}$ (6)

${V}_{nm}^{\left(\omega \right)}=\frac{3}{4\text{π}\left(n+3\right)\left(2n+1\right)}\frac{{{R}^{\prime }}^{\left(\omega \right)n+3}}{{R}^{n+3}}\underset{k=0}{\overset{\sigma }{\sum }}\left(\begin{array}{c}n+3\\ k\end{array}\right){\int }_{\sigma }\frac{{\rho }^{\left(\omega \right)}\left({\Omega }^{\prime }\right)}{\stackrel{¯}{\rho }}{F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right){Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (7)

${\rho }^{\left(\omega \right)}\left({r}^{\prime },{\Omega }^{\prime }\right)={h}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)\underset{i=0}{\overset{M}{\sum }}\left[{\alpha }_{i0}+{\alpha }_{i1}\left({R}^{\prime }-{r}^{\prime }\right)\right]{\chi }_{i}$ (8)

${V}_{nm}^{\left(\omega \right)}=\frac{3{h}^{\left(\omega \right)}}{4\text{π}\left(2n+1\right)}\frac{{{R}^{\prime }}^{\left(\omega \right)n+3}}{{R}^{n+3}}\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left[\frac{{\alpha }_{i0}+{\alpha }_{i1}{R}^{\prime }}{\stackrel{¯}{\rho }\left(n+3\right)}\underset{k=0}{\overset{\sigma }{\sum }}\left(\begin{array}{c}n+3\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)$ $-\frac{{\alpha }_{i1}{{R}^{\prime }}^{\left(\omega \right)}}{\stackrel{¯}{\rho }\left(n+4\right)}\underset{k=0}{\overset{\sigma +1}{\sum }}\left(\begin{array}{c}n+4\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)\right]{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (9)

2.3. 重力正演模型的效率与精度

Table 1. Density polynomials for Geological provinces, taking the upper crust boundary as the reference surface

$ϵ\sigma ={{R}^{\prime }}^{\left(\omega \right)n+3}\left[\frac{{\alpha }_{i0}+{\alpha }_{i1}{R}^{\prime }}{\stackrel{¯}{\rho }\left(n+3\right)}\underset{k=0}{\overset{\sigma }{\sum }}\left(\begin{array}{c}n+3\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)-\frac{{\alpha }_{i1}{{R}^{\prime }}^{\left(\omega \right)}}{\stackrel{¯}{\rho }\left(n+4\right)}\underset{k=0}{\overset{\sigma +1}{\sum }}\left(\begin{array}{c}n+4\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)\right]$ (10)

${V}_{nm}^{\left(\omega ,k\right)}=\frac{3{h}^{\left(\omega \right)}}{4\text{π}\left(2n+1\right)}\frac{{{R}^{\prime }}^{\left(\omega \right)n+3}}{{R}^{n+3}}\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left[\frac{{\alpha }_{i0}+{\alpha }_{i1}{R}^{\prime }}{\stackrel{¯}{\rho }\left(n+3\right)}\left(\begin{array}{c}n+3\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)-\frac{{\alpha }_{i1}{{R}^{\prime }}^{\left(\omega \right)}}{\stackrel{¯}{\rho }\left(n+4\right)}\left(\begin{array}{c}n+4\\ k\end{array}\right){F}_{k}^{\left(\omega \right)}\left({\Omega }^{\prime }\right)\right]{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (11)

3. 结果

Figure 1. Degree variances at different power under 3-D isotropic FSM of different geological provinces

4. 讨论

Table 2. Statistics of discrepancy between the spatial result of individual binominal expansion of each geological province and the overall resultant forward modelling gravity

Continued

5. 结论

${V}_{nm}^{\left(\omega ,1\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left[\frac{{\alpha }_{i0}+{\alpha }_{i1}\left({R}^{\prime }-{{R}^{\prime }}^{\left(\omega \right)}\right)}{\stackrel{¯}{\rho }}{F}_{1}^{\text{*}\left(\omega \right)}\left({\Omega }^{\prime }\right)\right]{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (1)

${V}_{nm}^{\left(\omega ,2\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right){R}^{\prime }-\left(n+3\right){{R}^{\prime }}^{\left(\omega \right)}\right]}{2\stackrel{¯}{\rho }}{F}_{2}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (2)

${V}_{nm}^{\left(\omega ,3\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right){R}^{\prime }-\left(n+3\right)\left(n+2\right){{R}^{\prime }}^{\left(\omega \right)}\right]}{6\stackrel{¯}{\rho }}{F}_{3}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (3)

${V}_{nm}^{\left(\omega ,4\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n{R}^{\prime }-\left(n+3\right)\left(n+2\right)\left(n+1\right){{R}^{\prime }}^{\left(\omega \right)}\right]}{24\stackrel{¯}{\rho }}{F}_{4}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (4)

${V}_{nm}^{\left(\omega ,5\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right){R}^{\prime }-\left(n+3\right)\left(n+2\right)\left(n+1\right)n{{R}^{\prime }}^{\left(\omega \right)}\right]}{120\stackrel{¯}{\rho }}{F}_{5}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (5)

${V}_{nm}^{\left(\omega ,6\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{\begin{array}{l}{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right){R}^{\prime }\\ -\left(n+3\right)\left(n+2\right)\left(n+1\right)n\left(n-1\right){{R}^{\prime }}^{\left(\omega \right)}\right]\end{array}}{720\stackrel{¯}{\rho }}{F}_{6}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (6)

${V}_{nm}^{\left(\omega ,7\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{\begin{array}{l}{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right){R}^{\prime }\\ -\left(n+3\right)\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)n{{R}^{\prime }}^{\left(\omega \right)}\right]\end{array}}{5040\stackrel{¯}{\rho }}{F}_{7}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (7)

${V}_{nm}^{\left(\omega ,8\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{\begin{array}{l}{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right){R}^{\prime }\\ -\left(n+3\right)\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right){{R}^{\prime }}^{\left(\omega \right)}\right]\end{array}}{40320\stackrel{¯}{\rho }}{F}_{8}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (8)

${V}_{nm}^{\left(\omega ,9\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{\begin{array}{l}{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)\left(n-5\right){R}^{\prime }\\ -\left(n+3\right)\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right){{R}^{\prime }}^{\left(\omega \right)}\right]\end{array}}{\text{362880}\stackrel{¯}{\rho }}{F}_{9}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (9)

${V}_{nm}^{\left(\omega ,10\right)}=\underset{i=0}{\overset{M}{\sum }}{\chi }_{i}{\int }_{\sigma }\left\{\frac{\begin{array}{l}{\alpha }_{i0}+{\alpha }_{i1}\left[\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)\left(n-5\right)\left(n-6\right){R}^{\prime }\\ -\left(n+3\right)\left(n+2\right)\left(n+1\right)n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)\left(n-5\right){{R}^{\prime }}^{\left(\omega \right)}\right]\end{array}}{\text{3628800}\stackrel{¯}{\rho }}{F}_{10}^{\text{*}\left(\omega \right)}\right\}{Y}_{nm}^{*}\left({\Omega }^{\prime }\right)\text{d}\Omega$ (10)

NOTES

*通讯作者。

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